First-principles calculations of diffusion coefficients in magnetic systems: Ni, Cr, and Ni-X alloys

Chelsey Zacherl, Shun-Li Shang, and Zi-Kui LiuNIST Diffusion WorkshopMay 4th, 2012

Acknowledgement: Office of Naval Research (ONR) under contract No. N0014-07-1-0638 and the Center for Computational Materials Design (CCMD), a joint National Science Foundation (NSF) Industry/University Cooperative Research Center at Penn State (IIP-1034965) and Georgia Tech (IIP-1034968).

Outline

• Current challenges in diffusion by first-principles

• Ni self-diffusion (fcc)– Ferromagnetic calculations

• Cr self-diffusion (bcc)– Anti-ferromagnetic calculations

• Impurity diffusion in Ni-X binary alloys– 26 Ni-X systems

• Non-dilute diffusion in Ni-Al fcc system• Summary and future work

2

Current challenges in calculating diffusion by first-principles• Vibrational properties at the saddle point

– SOLUTION: Eyring’s reaction rate theory and Nudged-Elastic Band (NEB) method

• Experimental self-diffusion data for Ni, Cr exists above magnetic transition temperature– Calculations performed with magnetic ordering

• Effect of magnetic ordering and disordering such as in Ni and Ni alloys– Empirical magnetic terms– Partition function/quantum Monte Carlo approach

3

Self-diffusion in cubic systems

• Vacancy-mediated

• In Arrhenius form:

wCfaD V2

f – jump correlation factora2 – lattice parameterC – vacancy concentrationw – atom jump frequency

)/exp(0 TkQDD BD0 – diffusion prefactor (intercept)Q – activation energy (slope)kB – Boltzmann's constant

4Mehrer, Diffusion in Solids, 2007

Initial vacancy configuration

Vacancy mediated self-diffusion in fcc Ni: calculating the least energy diffusion path within Eyring’s reaction rate theory

Saddle configuration

5Mantina, PSU Dissertation, 2008

Ni

1

3

2

Eyring, J. Chem. Phys. 1935

TkHexp

kSexp

hTkw

B

m

B

mB

TkH

expk

SexpC

B

f

B

fV

Vacancy formation and migration:

Fcc Ni self-diffusion coefficients calculation input details

• Vienna ab-inito Simulation Package (VASP)• Automated Theoretical Alloy Toolkit (ATAT) or Debye-

Grüneisen model for finite temperature vibrational contribution

• 32-atom Ni supercell (2 x 2 x 2)• Ferromagnetic spin on all atoms• PAW-LDA • Full relaxation of perfect and equilibrium vacancy

configurations• Nudged-Elastic Band (NEB) method for saddle point

configurations• No surface corrections used 6

Underestimation of diffusion parameters compared to experiments is a consistent trend in the present work

Zacherl, Acta Mat., Under review, 2012 7

QHA Debye model approach yields better agreement with experiments than QHA phonon method

8

**Filled data points:

Single-crystal

**Open data points:

Poly-crystal

Zacherl, Acta Mat., Under review, 2012

Tabulated data yields same results regarding QHA Debye vs QHA Phonon

Method References D0 (m2/sec) Q (eV) T (K)

DFT LDA QHA Debye (FM)

The present work

1.99 x 10-5 –2.14 x 10-4

2.85 – 3.03 700 - 1700

DFT LDA Debye QHA (NM)

The present work

5.27 x 10-6 – 3.92x 10-5

2.64 – 2.69 700 - 1700

DFT LDA QHA Phonon (FM)

The present work

3.90 x 10-6 –8.98 x 10-6

2.80 - 2.79 700 - 1700

DFT LDA HA Phonon (NM)

Mantina 7.00 x 10-6 –7.09 x 10-6

2.65 – 2.66 700 - 1700

Expt. (SC) Bakker 1.77 x 10-4 2.995 1253 - 1670Expt. (SC) Vladimirov 0.85 x 10-4 2.87 1326 - 1673

Expt. (SC) Maier 1.33 x 10-4 2.91 815 - 1195

9Zacherl, Acta Mat., Under review, 2012

Non-magnetic calculations show worse results than ferromagnetic calculations for both methods

10

Debye Phonon

Campbell, Acta Materialia, 2011

Thermodynamic properties of QHA FM LDA Debye compare well with experimental data

11

Ferromagnetic

Nonmagnetic

Zacherl, Acta Mat., Under review, 2012

Magnetization charge density shows mag-netic influence on thermodynamic properties

12

High

Low

Perfect Initial vacancy (corners) Saddle configuration

Looking down b axis(spin up – spin down)

Zacherl, Acta Mat., Under review, 2012

Diffusion of bcc AFM Cr shows good agreement with single-crystal data

13

**Filled data points:

Single-crystal

**Open data points:

Poly-crystal

Arrhenius parameters of Cr self-diffusion

Method References D0 * 104

(m2/sec)

Q (eV) T (K)

DFT PBE HA Phonon

The present work

41.20 4.05 1000

145.8 4.11 2000

Expt: Single crystal

Mundy, 1976 970 4.51 1369-2093

Expt: Single crystal

Mundy, 1981 40 4.58 1073-1446

14

Conclusions from self-diffusion work in magnetic systems• Magnetic effects are far reaching throughout

supercell and affect thermodynamic properties

• Overestimation of energy of vacancy regardless of X-C functional used– Surface effect magnified by magnetic properties

of fcc Ni

• Empiricism of the Debye model makes up for magnetic effects phonon is not representing– “double correction”

15

Dilute impurity diffusion in fcc ferromagnetic Ni-X alloys

* Experimental dilute impurity diffusion data

16

Approach: impurity diffusion in Ni-X determined from jump frequency of the impurity atom, w2

• LeClaire and Lidiard’s five frequency model

17LeClaire, Phil. Mag., 1956Mehrer, Diffusion in Solids, 2007

222

22 wCafD

3

4

0

2

0

2

0

2

ww

ww

ff

DD

Impurity correlation factor

Validation of methodology by comparing Q, activation barrier for diffusion to previous 0 K study

Janotti, Physical Review Letters, 2004 18

Test system: LDA QHA Debye model for vibrational contribution, no electronic used

**Filled data points:

Single-crystal

**Open data points:

Poly-crystal

Ni-Cu

19

Arrhenius parameters including temperature dependence at all temperatures

D0 (10-4m2/sec) Q (eV) T (K)

This work – LDA Debye

0.004 2.58 700

0.088 2.58 1700

Expt., Monma, 1964

0.57 2.676 1327 – 1632

Expt., Anand, 1965 0.66 2.645 1123 – 1323

Expt., Helfmeier, 1970

0.27 2.646 1048 – 1323

Expt, Taguchi, 1984

0.61 2.641 1080 – 1613

20

Results from selected impurity diffusion coefficient calculations

Ni-Al Ni-Ti

**Filled data points: **Open data points: Single-crystal Poly-crystal

21

Results from selected impurity diffusion coefficient calculations

22

**Filled data points: **Open data points: Single-crystal Poly-crystal

Ni-FeNi-Cr

Results from selected impurity diffusion coefficient calculations

**Filled data points: **Open data points: Single-crystal Poly-crystal

23

Comments on the Ni-X impurity diffusion results

• Underestimation of one order of magnitude compared to experimental data is consistent for all systems shown– Affecting activation barrier for diffusion, Q and

diffusion prefactor, D0

– Partially attributed to lack of complete relaxation of the three transition states in VASP

• Lingering question:– How to analyze data from 26 systems– Charge density

24

Full relaxation in VASP 5 using ss-CINEB TST tools from UT Austin shows slight correction at lower temperatures

**Filled data points:

Single-crystal

**Open data points:

Poly-crystal

25Henkelman, Chem. Phys. 2000

How can we provide understanding of the effects of each X on the impurity diffusion coefficient?

26

Relative charge density analysis of Ni-Re compared to that of pure Ni

27

Pure Ni: Saddle configuration

High

Low

Pure Ni: Saddle configuration

a axis c axisb axis

(spin up – spin down)

Relative charge density analysis of Ni-Y compared to that of pure Ni

28

Pure Ni: Saddle configuration

High

Low

Pure Ni: Saddle configuration

a axis c axisb axis

(spin up – spin down)

Additional challenge: non-dilute impurity diffusion of Al in fcc FM Ni

• Bocquet’s 14 frequency model to predict all possible vacancy interactions with lone or paired solute atoms

• LeClaire’s 5 frequency model for lone solute atom – vacancy interactions

Bocquet, Report: CEA-R-4292, 1972Le Claire, J. Nuclear Mater. 69-70(1978) 70

Open circle: solventFilled circle: soluteSquare: vacancy

29

14 Frequency model• Calculations completed for Ni-Al system

– 30 Ni atoms and 2 Al atoms– Solvent enhancement factor, b1

– Solute enhancement factor, B1

– Isotope effect measurement, E

• Current progress:– Empirical relation for impurity diffusion coefficient

dependent on concentration:

LeClaire, Journal of Nuclear Materials, 1978 30

• Impurity diffusion coefficient for 26 FM Ni-X alloy systems– Deformation charge density– Trends in data

• Future: 14-frequency model– Fit to Redlich-Keister polynomial

• All calculations: limitations of magnetic model in comparison to experimental data– Did not include transition effects; shown to be less for

Ni than other elements like Co or Fe– Debye vs. Phonon methods

Conclusions and future work for this project

31Shang, J. Appl. Phys., 2010 Kormann, PRB, 2011

Ni Fe

Thank you!

32

Phonon calculations

• Based on the partition function of lattice vibrations, we have the phonon frequency as a function, g(w)

www dgTk

TkTVFB

Bvib

0 2sinh2ln,

33Shang, Acta. Mat., 2005Wang, Acta. Mat., 2004

Phonon frequency Phonon density of states

Debye-Grüneisen Model• Speed of sound is constant in a material• Linear density of states

34

Debye function:

Debye - Grüneisen model for finding q(D) and solving above equation:

Shang, Comp. Mat. Sci., 2010Moruzzi, Phys. Rev. B., 1988

Zero-point energy from lattice vibrations

Grüneisen constant

35

Eyring’s Reaction Rate Theory

• Partition function • Atom jump frequency

TkHH

kSS

hTkw

B

NeqNsd

B

NeqNsdB 1,*

1,1,*

1, expexp

eq

sdB

ZZ

hTkw

*

* remove the imaginary frequency of

the saddle configuration

T

SN

NS

Tk

HN

NHzfrCwfaD

NNsd

B

NNsd1

exp

1

exp61

*1,

*1,

22

TkGZ

B

exp

Eyring, J. Chem. Phys., Vol.3, 1935, 107 35

Fcc Ni self-diffusion coefficients calculation input details

• Vienna ab-inito Simulation Package (VASP)• Automated Theoretical Alloy Toolkit (ATAT) or

Debye-Grüneisen model for vibrational contribution• 32-atom Ni supercell (2 x 2 x 2)• Ferromagnetic spin on all atoms• PAW-LDA or PAW-PBEsol• Full relaxation of perfect, and equilibrium vacancy

configurations• Nudged-Elastic Band (NEB) method for saddle

point configurations36

Diffusion coefficient equations

)/exp(0 TkQDD B

B

PSB

k

SSfa

hTkD SC 32

31exp

*2

0 PSSC HHQ 3231*

wCfaD V2

Mantina, Defect and Diffusion Forum, 2009

Ni NM diffusion coefficient by harmonic phonon calculations

Mantina, PhD. Thesis, Penn State University, 2008

bcc Cr self-diffusion coefficients calculation input details

• Vienna ab-inito Simulation Package (VASP)• Debye-Grüneisen model for vibrational contribution• 54-atom Cr supercell (2-atom bcc base)

– Spin of +/- 1.06 mB/atom

• Anti-ferromagnetic spin• PAW-GGA-PBE• Full relaxation of perfect and equilibrium vacancy

configurations• Nudged-Elastic Band (NEB) method for saddle point

configurations– Full relaxation

39

Impurity diffusion coefficient calculation input details

• Vienna ab-inito Simulation Package (VASP)• 32 atom, supercell, N-2 Ni atoms, 1 impurity, 1

vacancy 8 total configurations• Ferromagnetic spin on all atoms• PAW-LDA• Full relaxation of perfect, and equilibrium vacancy

configurations• NEB or Climbing Image Nudged-Elastic Band

(CINEB) method for saddle point configurations– Relax only ionic positions

• Debye model for vibrational contribution 40

Impurity diffusion coefficient determined from the jump frequency of the impurity atom, w2

• Equation for impurity diffusion is related to tracer diffusion:

• With vacancy concentration and jump frequency:

• And the binding energy with impurity, X, and vacancy, V:

LeClaire, J. Nuclear Matls, 1978Wolverton, Acta Mat., 2007

222

22 wCafD

TkGG

CB

bfexp2

)()()()()( 1111112 VNiGXNiGNiGVXNiGVXG NNNNb

41

TkGG

hTkw

B

wIS

wTSB

22

exp2

3

4

0

2

0

2

0

2

ww

ww

ff

DD

Calculation of the impurity correlation factor, f2, involves calculating all five jump frequencies

• f2 is related to the probability of the impurity atom making the reverse jump back to its previous position

• Where F is the “escape probability” that after a dissociation jump, the vacancy will not return to a first nearest neighbor site of the impurity

Mehrer, Diffusion in Solids, 2007Manning, Physical Review, 1964

x = w4/w0

42

Impurity diffusion coefficient for Ni-Si and Ni-Ti systems

Ni-Si Ni-Ti

**Filled data points: **Open data points: Single-crystal Poly-crystal

43

Impurity diffusion coefficient for Ni-Nb and Ni-Zr systems

**Filled data points: **Open data points: Single-crystal Poly-crystal

Ni-Nb

Ni-Zr

44

Impurity diffusion coefficient for Ni-W and Ni-V systems

**Filled data points: **Open data points: Single-crystal Poly-crystal

Ni-W Ni-V

45

Other impurity elements

3d transition elements in Ni

47

4d transition elements in Ni

48

5d transition elements in Ni

49

LeClaire’s 5 freqnecies• (1) w0: solvent–vacancy exchange frequency (far

from a solute atom).• (2) w1: solvent–vacancy exchange frequency for the

rotational jump around one single solute atom• (3) w3: solvent–vacancy exchange frequency for the

solute–vacancy dissociative jump• (4) w4: solvent–vacancy exchange frequency for the

solute–vacancy associative jump• (5) w2: single solute–vacancy exchange frequency

Bocquet’s 14 frequencies• (6) w33: solvent–vacancy exchange frequency for the jump that takes

the vacancy away from the solute pair• (7) w34: solvent–vacancy exchange frequency for the jump that takes

the vacancy away from one solute atom in the single atom configuration but at the same time brings the vacancy into the nearest neighbor position for the other solute atom

• (8) w44: solvent–vacancy exchange frequency for the jump that brings the vacancy to the nearest neighbor position for the solute pair

• (9) w14: solvent–vacancy exchange frequency for the jump that brings the vacancy from the nearest neighbor position to one of the solute atoms in the pair configuration to the nearest neighbor position of both solute atoms in the pair configuration.

Bocquet’s 14 frequencies, con’t• (10) w13: solvent–vacancy exchange frequency for the jump that takes the vacancy

from the nearest neighbor position to both solute atoms in the pair configuration to the nearest neighbor position to one of the solute atoms in the pair configuration.

• (11) w11: solvent–vacancy exchange frequency for the jump that keeps the vacancy at the nearest neighbor position to both solute atoms in the pair configuration.

• (12) w23: solute–vacancy exchange frequency for the jump that takes a solute atom away from the other solute

• (13) w21: solute–vacancy exchange frequency for a solute rotational jump around the other solute atom

• (14) w24: solute–vacancy exchange frequency for the jump that brings a solute atom to the nearest neighbor position of the other solute atom